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<p>(ii) Sufficient ConditionSuppose that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_26.html ./knowl/eq2_25.html">
\begin{equation*}
\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}~\textrm{in} ~ R.
\end{equation*}
</div>
<p class="continuation">Next, we consider a function <span class="process-math">\(\Psi(x, y)\)</span> defined by</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_26.html ./knowl/eq2_25.html">
\begin{equation*}
\Psi(x, y)=\int M(x, y) \textrm{d} x+\int \left[N(x, y) -\int \frac{\partial M(x, y)}{\partial y} \textrm{d} x  \right] \textrm{d} y.
\end{equation*}
</div>
<p class="continuation">Then we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_26.html ./knowl/eq2_25.html">
\begin{equation*}
\begin{aligned}
&amp;\frac{\partial \Psi}{\partial x}=M(x, y)+\int \left[\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y} \right] \textrm{d} y=M(x, y),\\
&amp; \frac{\partial \Psi}{\partial y}=\int \frac{\partial M}{\partial y} \textrm{d} x  +N(x, y)-\int \frac{\partial M}{\partial y} \textrm{d}x=N(x, y)     .
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Thus, for such a <span class="process-math">\(\Psi(x, y)\text{,}\)</span> (<a href="" class="xref" data-knowl="./knowl/eq2_26.html" title="Equation 2.6.2">(2.6.2)</a>) is satisfied, i. e., (<a href="" class="xref" data-knowl="./knowl/eq2_25.html" title="Equation 2.6.1">(2.6.1)</a>) is an exact ODE.</p>
<span class="incontext"><a href="sec2_6.html#p-46" class="internal">in-context</a></span>
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